Optimal. Leaf size=65 \[ \frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{4 a} \]
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Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4836, 4473, 2234, 2204} \[ \frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 4473
Rule 4836
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2} e^{-i x+x^2}+\frac {1}{2} e^{i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac {\operatorname {Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt [4]{e} \operatorname {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac {\sqrt [4]{e} \operatorname {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-i+2 \sin ^{-1}(a x)\right )\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (i+2 \sin ^{-1}(a x)\right )\right )}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.74 \[ \frac {\sqrt [4]{e} \sqrt {\pi } \left (\text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )+\text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )\right )}{4 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (\arcsin \left (a x\right )^{2}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsin \left (a x \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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